If the perimeter of a circular sector is fixed at 100 ft, what values of r and s give the sector the greatest area?

The sector arc length is just a fraction of the circumference: 
Laʀc = (2πR) • (θ ⁄ 2π) = R • θ ... where θ = central angle (of sector) 

 Ps = R + R + (R • θ) ... perimeter of sector = Ps 

 Ps = R • (2 + θ) ... Ps = 100 ft 

  100 = R • (2 + θ) 

      θ = (100 ⁄ R) – 2 

... 

Similarly, the sector area is just a fraction of the circle area: 

 As = (πR²) • (θ ⁄ 2π) ... area of sector = As 

 As = R² • θ ⁄ 2 ... substitute for θ 

 As = R² • [(100 ⁄ R) – 2 ] ⁄ 2 

 As = 50R – R² ... differentiate 

            As' = 50 – 2R ... set to zero 

           0 = 50 – 2R 

         R = 25 ft ... optimum radius 

... 

 As = 50R – R² ... evaluate at: R = 25 ft 

 As = 50(25) – (25)² 

 As = 625 ft² ... maximum area 

Note ... at any other R_value, the sector area is less. 

  " θ " can be determined using:  θ = (100 ⁄ R) – 2



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The mode of the numbers is 29.